Let us start by clarifying this a bit. I am aware of some proofs that irrationals/reals are uncountable. My issue comes by way of some properties of the reals. These issues can be summed up by the combination of the following questions:

- Is it true that between any two rationals one may find at least one irrational?
- Is it true that between any two irrationals one may find at least one rational?
- Why are the reals uncountable?

I've been talking with a friend about why the answer of these three questions can be the case when they somewhat seem to contradict each other. I seek clarification on the subject. Herein lies a summary of the discussion:

**Person A:** *By way of Cantor Diagonalization it can be shown that the reals are uncountable.*

**Person B:** *But is it not also the case that one may find at least one rational between any two irrationals and vice versa?*

**Person A:** *That seems logical, I can't pose a counterexample... but why does that matter?*

**Person B:** *Wouldn't that imply that for every irrational there is a corresponding rational? And from this the Reals would be equivalent to 2 elements for every element of the rationals?*

**Person A:** *That implies that the Reals are countable, but we have already shown that they weren't... where is the hole in our reasoning?*

**And so I pose it to you... where is the hole in our reasoning?**